On universal covers in normed planes

Abstract

Abstract In 1914, Lebesgue posed his famous universal cover problem for the Euclidean plane which is still unsolved. We present the first explicit investigation of this problem for arbitrary normed planes. A convex body K in a finite dimensional real Banach space is called universal cover if any set of diameter 1 can be covered by a congruent copy of K; if ”congruent copy“ is replaced by ”translate”, then K is called strong universal cover. We describe the smallest regular hexagons, the smallest equilateral triangles, and the smallest squares (all these figures defined in the sense of the norm) that are strong universal covers in normed planes. In addition, the paper contains a new characterization of Radon planes. In view of universal covers having ball shape we shortly discuss also Jung constants of real Banach spaces, and a known result on Borsuk’s partition problem is reproved.